Liberating the weights

A partial answer to why quasi-Monte Carlo (QMC) algorithms work well for multivariate integration was given in Sloan and Woźniakowski (J. Complexity 14 (1998) 1-33) by introducing weighted spaces. In these spaces the importance of successive coordinate directions is quantified by a sequence of weights. However, to be able to make use of weighted spaces for a particular application one has to make a choice of the weights. In this work, we take a more general view of the weights by allowing them to depend arbitrarily not only on the coordinates but also on the number of variables. Liberating the weights in this way allows us to give a recommendation for how to choose the weights in practice. This recommendation results from choosing the weights so as to minimize the error bound. We also consider how best to choose the underlying weighted Sobolev space within which to carry out the analysis. We revisit also lower bounds on the worst-case error, which change in many minor ways now, since the weights are allowed to depend on the number of variables, and we do not assume that the weights are uniformly bounded as has been assumed in previous papers. Necessary and sufficient conditions for QMC tractability and strong QMC tractability are obtained for the weighted Sobolev spaces with general weights. In the final section, we show that the analysis of variance decomposition of functions from one of the Sobolev spaces is equivalent to the decomposition of functions with respect to an orthogonal decomposition of this space.

[1]  F. J. Hickernell,et al.  Tractability of Multivariate Integration for Periodic Functions , 2001, J. Complex..

[2]  S. C. Zaremba.  Some applications of multidimensional integration by parts , 1968 .

[3]  Ian H. Sloan,et al.  Why Are High-Dimensional Finance Problems Often of Low Effective Dimension? , 2005, SIAM J. Sci. Comput..

[4]  Joseph F. Traub,et al.  Faster Valuation of Financial Derivatives , 1995 .

[5]  B. Efron,et al.  The Jackknife Estimate of Variance , 1981 .

[6]  H. Wozniakowski.  Efficiency of Quasi-Monte Carlo Algorithms for High Dimensional Integrals , 2000 .

[7]  Henryk Wozniakowski,et al.  Intractability Results for Integration and Discrepancy , 2001, J. Complex..

[8]  Gerhard Larcher,et al.  On the tractability of the Brownian Bridge algorithm , 2003, J. Complex..

[9]  Kai-Tai Fang,et al.  The effective dimension and quasi-Monte Carlo integration , 2003, J. Complex..

[10]  Frances Y. Kuo,et al.  Reducing the construction cost of the component-by-component construction of good lattice rules , 2004, Math. Comput..

[11]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[12]  Frances Y. Kuo,et al.  Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces , 2003, J. Complex..

[13]  Henryk Wozniakowski,et al.  Tractability of Multivariate Integration for Weighted Korobov Classes , 2001, J. Complex..

[14]  I. H. SLOAN,et al.  Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces , 2002, SIAM J. Numer. Anal..

[15]  Henryk Wozniakowski,et al.  Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces , 2002, J. Complex..

[16]  A. Owen,et al.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 1997 .

[17]  Henryk Wozniakowski,et al.  Weighted Tensor Product Algorithms for Linear Multivariate Problems , 1999, J. Complex..

[18]  Christine Thomas-Agnan,et al.  Computing a family of reproducing kernels for statistical applications , 1996, Numerical Algorithms.

[19]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[20]  Frances Y. Kuo,et al.  Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points , 2002, J. Complex..

[21]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[22]  Frances Y. Kuo,et al.  On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces , 2002, Math. Comput..